We consider class of directed graphs (self edges (=loop) are allowed). The aim is: construct example of two graphs $G_1$ and $G_2$ such that:
- for each graph $H$ with $\le 7$ nodes $G_1$ contains induced subgraph isomorphic to $H$ only and only if $G_2$ contains induced subgraph isomorphic to $H$.
- Player $1$ (spojler) has winning strategy in Game with $7$ rounds.
To my eye, is is sufficient to give $K_8$ and $K_7$, but each node has loop (edge to self).
For the first point, we know that $K_8$ contains as subgraph $K_7$, so if $K_9$ has
subgraph isomorphic to some $H=(V,_)$ such that $|V|\le 7$ so $K_8$ also.
If $K_8$ has subgraph isomorphic to some $H$ then we may say the same about $K_7$ because $H$ has at most $7$ nodes, we may close this $H$ in subgraph of $K_8$ which is simultatenously $K_7$.
For the second point. We know that in our $K_8$ each node has exactly $7$ distinct neighbours, however in $K_7$ each node has only $6$ distinct neighbours. So, after $7$ moves of player $1$ duplicator has no move. Simlpy, Player one choose any node of $K_8$ and then choose all its neighbours.
Tell me please, is it correct ? If not, how to fix it ?